- #1

Mystic998

- 206

- 0

## Homework Statement

For prime p, nonzero [itex]a \in \bold{F}_p[/itex], prove that [itex]q(x) = x^p - x + a[/itex] is irreducible over [itex]\bold{F}_p[/itex].

## Homework Equations

## The Attempt at a Solution

It's pretty clear that none of the elements of [itex]\bold{F}_p[/itex] are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if [itex] \alpha[/itex] is a root of q(x), then so is [itex]\alpha + 1[/itex], and from there I was able to deduce (hopefully not incorrectly) that [itex]\bold{F}_{p}(\alpha)[/itex] is the splitting field for q(x) over the field, but I can't figure out the degree of the extension, so I'm kind of stuck. Any ideas?